Beavers and otters. They nearly hunted them to extinction in Europe, the pelts of these animals have far superior hair density and the fur strongly repels water, perfect for rain/water proof hats and coats. If your a rich aristocrat in Europe you don't have to settle for non-waterproof clothing you can buy the good stuff and walk out into the rain without getting sopping wet like a peasant. Also foxes and several types of weasels have soft and very attractive fur for making those crazy fur capes kinds and queens wore, and were now becoming all the rage with the up and coming middle class. By now Europe was extremely over exploited for such fine furs and there was this content where they were barely exploited with only a very small home market so they were all exported to feed fashion crazes in Europe. Like such things as mink coats and top hats made from beaver pelts, there was a time for a about a century where every man that thought he was anything had one.

This might not count as "the early days of colonization" in North America, but in the late 18th century and into the early 19th there was a huge market for furs in China. For a while the Chinese government, which was suspicious of foreign trade, kept the trade under tight controls. British and American fur traders had to trade through Canton, while Russians were required to trade furs via Kyakhta, in what is now Mongolia. The tight controls played a role in driving up the price. As for why there was a large demand for furs in China, historian James Gibson writes:

Fuel was scarce and costly in China, and in North China winters were long and cold. Even at Canton "all the Chinese that can afford it" wore woollens or fur-lined camlets during the cool season of March-April, when frost could occur at night. And South China's houses were unheated. An early British Nor'westman, Captain John Meares, noted that "even there [Canton Province], the cold will often render a fur dress necessary [...]". The anonymous editor of a Cantonese English-language annual stated that the consumption of furs was "very great" because "the necessity of restricting the use of fuel to culinary operations and the arts, compels the Chinese to load themselves with garments in the winter." So their winter clothing was lined with fur or stuffed with cotton in order to limit the number of outer garments; fur-lined clothes were carefully preserved and often passed from father to son, usually lasting about twelve years.

He goes on:

Sea-otter was preferred in North China and beaver in South China. Captain Meares explained: "The skin of the sea-otter, from the thickness of its pile and the length of its fur, forms too cumbersome an habiliment for the people of the Southern provinces; they prefer, in general, the Canadian and Hudson's Bay [beaver] furs; but still, such as can afford it, seldom fail of having a cape of the sea-otter's skin to their coats, though perhaps at the extravagant price of six dollars." [...] And the demand for furs at Canton was "steady and constant". Captain Meares believed that all of Canton's intake of sea-otter skins before 1790 had not even satisfied the demand of Canton Province alone.

During the trading season in Canton buyers would come from as far as North China with teas, silks, and ivory to trade for furs and cloth. Fur traders in Canton sold furs in exchange for lots of tea, silk, nankeen, crape, porcelains (to the point where porcelain dinnerware and the like became known as chinaware or just china), sugar, cinnamon, teak, and much else. Profit was made selling these things in Europe and America. In fact, the most successful American fur traders operated what was known as "the golden round"—a circumnavigation of the world with profit to be made three times over. First cheap "trade goods", like steel tools, sheets of copper, and such, were traded on the Pacific Northwest Coast for furs, at a profit. Then the furs were sold in Canton at a profit. Then Chinese goods were sold in Europe or America at a profit.

James Gibson argues that between about 1790 and 1840 this Northwest-China trade brought in so much money to New England, where most of the American maritime fur traders were based, that it played a key role in lifting New England out of a post-Revolutionary War depression and driving the region's transformation from an agrarian society to an industrial one.

Of course the early profits made encouraged more and more traders to enter the market, competing with one another and increasing the risk. At the same time the sea-otter population declined rapidly, with regional extinction and almost total extinction. As sea-otters became scarce their price rose, making ventures even more risky. By the 1830s and 40s the system began to collapse. Many maritime fur traders switching to sealing or whaling. The beaver trade remained profitable for a bit longer, but even by the 1830s the Hudson's Bay Company saw the writing on the wall and began diversifying into things like salmon and lumber.

I know less about earlier colonial times, but in the late 17th and early 18th centuries the deerskin trade was very important in the American southeast. In the late 17th century there was a fashion for beaver fur/felt in Europe, but this declined in the early 18th century. In London there was a shift in fashion toward leather hats which required deerskin. In addition, deerskin was important for making other leather goods, and the various European wars around the start of the 18th century drove demand for many leather goods. The deerskin trade boomed to such a degree that deer nearly became extinct in the American southeast.

Main source here: James R. Gibson, Otter Skins, Boston Ships, and China Goods. Also Richard Mackie, Trading Beyond the Mountains: The British Fur Trade on the Pacific 1793-1843. On the deerskin trade, William Ramsey, The Yamasee War: A Study of Culture, Economy, and Conflict in the Colonial South.